0.1 Case Description

Dos cursos de matemáticas de octavo básico, presentan diferencias importantes en sus resultados en una guia de trabajo. Estas guías, son aplicadas como preparación para las evaluación de cierre de semestre. Son versiones muy cortas de cómo lucen las pruebas finales.
El contenido de la guía aplicada incluía resolución de problemas (5 items), ejercicios de geometría (4 items), y 4 preguntas de algebra.
Ambos cursos son enseñados por el mismo profesor, pero en momentos diferentes de la semana. Un curso tiene clases los Lunes, y el otro curso tiene clases los Viernes. Sin embargo, las pruebas de cierre de trimestre y sus ensayos, son realizadas de forma sistemática los Miércoles.
Las diferencias entre las notas obtenidas por ambos cursos desaparecen, cuando las preguntas de algebra son ignoradas.
¿Quizas el curso del Viernes, no alcanzo a revisar los contenidos de algebra?

1 Cargar datos

1.1 Abrir datos

#--------------------------------------------------------------------
# load data
#--------------------------------------------------------------------

data_wide <- readRDS('rasch_bias.rds')

# ----------------------------------------------- 
# check data format
# -----------------------------------------------

library(dplyr)
r4sda::variables_table(data_wide) %>%
knitr::kable()

variable	type	values	labels
i01	dbl	, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, …	problem 1
i02	dbl	, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, …	problem 2
i03	dbl	, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, …	problem 3
i04	dbl	, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, …	problem 4
i05	dbl	, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, …	problem 5
i06	dbl	, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, …	geometry 1
i07	dbl	, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, …	geometry 2
i08	dbl	, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, …	geometry 3
i09	dbl	, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, …	geometry 4
i10	dbl	, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, …	algebra 1
i11	dbl	, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, …	algebra 2
i12	dbl	, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, …	algebra 3
i13	dbl	, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, …	algebra 4
u	dbl	, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, …	section (monday == 1, friday == 0)

1.2 Re-structure data

#--------------------------------------------------------------------
# prepare data
#--------------------------------------------------------------------

# ----------------------------------------------- 
# variables names
# -----------------------------------------------

data_stack <- data_wide %>%
              mutate(id_i = seq(1:nrow(.))) %>%
              tidyr::gather('item', 'resp', -u, -id_i) %>%
              dplyr::glimpse()

## Warning: attributes are not identical across measure variables;
## they will be dropped

## Rows: 1,040
## Columns: 4
## $ u    [3m[38;5;246m<dbl>[39m[23m 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1,…
## $ id_i [3m[38;5;246m<int>[39m[23m 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19…
## $ item [3m[38;5;246m<chr>[39m[23m "i01", "i01", "i01", "i01", "i01", "i01", "i01", "i01", "i01", "i…
## $ resp [3m[38;5;246m<dbl>[39m[23m 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1,…

2 Item bias with the logistic method

2.1 Item bias with `difR` with logistic method

#--------------------------------------------------------------------
# item bias
#--------------------------------------------------------------------

# ----------------------------------------------- 
# display results
# -----------------------------------------------

difR::difLogistic(
  Data = data_wide, 
  group = 'u', 
  focal.name = 0, 
  alpha = .05, 
  type = 'udif', 
  criterion = 'LRT'
  )

## 
## Detection of uniform Differential Item Functioning
## using Logistic regression method, without item purification
## and with LRT DIF statistic
## 
## Matching variable: test score 
##  
## No set of anchor items was provided 
##  
## No p-value adjustment for multiple comparisons 
##  
## Logistic regression DIF statistic: 
##  
##     Stat.  P-value   
## i01 4.0945 0.0430  * 
## i02 0.0348 0.8520    
## i03 0.6406 0.4235    
## i04 0.0052 0.9426    
## i05 1.9280 0.1650    
## i06 5.9003 0.0151  * 
## i07 0.0680 0.7943    
## i08 0.1173 0.7320    
## i09 2.0562 0.1516    
## i10 3.4581 0.0629  . 
## i11 1.5006 0.2206    
## i12 4.2026 0.0404  * 
## i13 9.8706 0.0017  **
## 
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1  
## 
## Detection threshold: 3.8 (significance level: 0.05)
## 
## Items detected as uniform DIF items:
##     
##  i01
##  i06
##  i12
##  i13
## 
##  
## Effect size (Nagelkerke's R^2): 
##  
## Effect size code: 
##  'A': negligible effect 
##  'B': moderate effect 
##  'C': large effect 
##  
##     R^2    ZT JG
## i01 0.0586 A  B 
## i02 0.0006 A  A 
## i03 0.0073 A  A 
## i04 0.0001 A  A 
## i05 0.0223 A  A 
## i06 0.0611 A  B 
## i07 0.0009 A  A 
## i08 0.0013 A  A 
## i09 0.0203 A  A 
## i10 0.0317 A  A 
## i11 0.0165 A  A 
## i12 0.0683 A  B 
## i13 0.1072 A  C 
## 
## Effect size codes: 
##  Zumbo & Thomas (ZT): 0 'A' 0.13 'B' 0.26 'C' 1 
##  Jodoin & Gierl (JG): 0 'A' 0.035 'B' 0.07 'C' 1 
## 
##  Output was not captured!

# Note: focal group is Friday, the gruoup without
#       without the oportunity to learn.


# ----------------------------------------------- 
# create dif object
# -----------------------------------------------

dif_results <- difR::difLogistic(
  Data = data_wide, 
  group = 'u', 
  focal.name = 0, 
  alpha = .05, 
  type = 'udif', 
  criterion = 'LRT'
  )

# ----------------------------------------------- 
# plot per item
# -----------------------------------------------

# problem items
plot(dif_results, plot = "itemCurve", item = 1, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 2, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 3, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 4, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 5, itemFit = "null")

## The plot was not captured!

# geometry items
plot(dif_results, plot = "itemCurve", item = 6, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 7, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 8, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 9, itemFit = "null")

## The plot was not captured!

# algebra items
plot(dif_results, plot = "itemCurve", item = 10, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 11, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 12, itemFit = "null")

## The plot was not captured!

plot(dif_results, plot = "itemCurve", item = 13, itemFit = "null")

## The plot was not captured!

2.2 Replicate statistics manually

#--------------------------------------------------------------------
# item bias
#--------------------------------------------------------------------

# ----------------------------------------------- 
# separate response matrix
# -----------------------------------------------

items_data <- dplyr::select(data_wide, i01:i13)

# ----------------------------------------------- 
# create total score
# -----------------------------------------------

data_dif <- data_wide %>%
            mutate(score = r4sda::mean_score(items_data)) %>%
            mutate(z_score = r4sda::z_score(score))

#-------------------------------------------------
# fit logistic method for uniform dif manually
#-------------------------------------------------

m0 <- glm(i13 ~ z_score, data = data_dif, family = "binomial")
m1 <- glm(i13 ~ z_score + u, data = data_dif, family = "binomial")

# ----------------------------------------------- 
# display results
# -----------------------------------------------

texreg::screenreg(list(m0, m1), 
    type = 'un',
    star.symbol = "*", 
    center = TRUE, 
    doctype = FALSE,
    dcolumn = TRUE, 
    booktabs = TRUE,
    single.row = TRUE)

## 
## ====================================================
##                 Model 1            Model 2          
## ----------------------------------------------------
## (Intercept)      -2.18 (0.54) ***   -3.06 (0.73) ***
## z_score           2.43 (0.58) ***    2.62 (0.68) ***
## u                                    2.68 (0.96) ** 
## ----------------------------------------------------
## AIC              56.16              48.29           
## BIC              60.92              55.43           
## Log Likelihood  -26.08             -21.14           
## Deviance         52.16              42.29           
## Num. obs.        80                 80              
## ====================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05

# ----------------------------------------------- 
# replicate lrt stat
# -----------------------------------------------

# get LRT for null model
d0 <- broom::glance(m0) %>%
dplyr::select(deviance) %>%
dplyr::pull()

# get LRT for model 1
d1 <- broom::glance(m1) %>%
dplyr::select(deviance) %>%
dplyr::pull()

# replicate stat
d0-d1

## [1] 9.9

# ----------------------------------------------- 
# replicate r2 effect size
# -----------------------------------------------

DescTools::PseudoR2(m1, "Nagel") - DescTools::PseudoR2(m0, "Nagel")

## Nagelkerke 
##       0.11

2.3 Item bias with anchors

#--------------------------------------------------------------------
# item bias
#--------------------------------------------------------------------

# ----------------------------------------------- 
# display results
# -----------------------------------------------

difR::difLogistic(
  Data = data_wide, 
  group = 'u', 
  focal.name = 0, 
  alpha = .05, 
  type = 'udif', 
  criterion = 'LRT'
  )

## 
## Detection of uniform Differential Item Functioning
## using Logistic regression method, without item purification
## and with LRT DIF statistic
## 
## Matching variable: test score 
##  
## No set of anchor items was provided 
##  
## No p-value adjustment for multiple comparisons 
##  
## Logistic regression DIF statistic: 
##  
##     Stat.  P-value   
## i01 4.0945 0.0430  * 
## i02 0.0348 0.8520    
## i03 0.6406 0.4235    
## i04 0.0052 0.9426    
## i05 1.9280 0.1650    
## i06 5.9003 0.0151  * 
## i07 0.0680 0.7943    
## i08 0.1173 0.7320    
## i09 2.0562 0.1516    
## i10 3.4581 0.0629  . 
## i11 1.5006 0.2206    
## i12 4.2026 0.0404  * 
## i13 9.8706 0.0017  **
## 
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1  
## 
## Detection threshold: 3.8 (significance level: 0.05)
## 
## Items detected as uniform DIF items:
##     
##  i01
##  i06
##  i12
##  i13
## 
##  
## Effect size (Nagelkerke's R^2): 
##  
## Effect size code: 
##  'A': negligible effect 
##  'B': moderate effect 
##  'C': large effect 
##  
##     R^2    ZT JG
## i01 0.0586 A  B 
## i02 0.0006 A  A 
## i03 0.0073 A  A 
## i04 0.0001 A  A 
## i05 0.0223 A  A 
## i06 0.0611 A  B 
## i07 0.0009 A  A 
## i08 0.0013 A  A 
## i09 0.0203 A  A 
## i10 0.0317 A  A 
## i11 0.0165 A  A 
## i12 0.0683 A  B 
## i13 0.1072 A  C 
## 
## Effect size codes: 
##  Zumbo & Thomas (ZT): 0 'A' 0.13 'B' 0.26 'C' 1 
##  Jodoin & Gierl (JG): 0 'A' 0.035 'B' 0.07 'C' 1 
## 
##  Output was not captured!

# Note: focal group is Friday, the gruoup without
#       without the oportunity to learn.


# ----------------------------------------------- 
# create dif object
# -----------------------------------------------

dif_anchored <- difR::difLogistic(
  Data = data_wide, 
  anchor = 1:9,
  group = 'u', 
  focal.name = 0, 
  alpha = .05, 
  type = 'udif', 
  criterion = 'LRT'
  )

# ----------------------------------------------- 
# plot per item
# -----------------------------------------------

# algebra items
plot(dif_anchored, plot = "itemCurve", item = 10, itemFit = "null")

## The plot was not captured!

plot(dif_anchored, plot = "itemCurve", item = 11, itemFit = "null")

## The plot was not captured!

plot(dif_anchored, plot = "itemCurve", item = 12, itemFit = "null")

## The plot was not captured!

plot(dif_anchored, plot = "itemCurve", item = 13, itemFit = "null")

## The plot was not captured!

3 Item bias as an special case of explanatory response model problem

3.1 Fit Rasch model as a mixed model

\[ln[Pr(y_{ip} = 1)] = \theta_{.p} - \delta_{1-13}item\]

#--------------------------------------------------------------------
# fit rasch model
#--------------------------------------------------------------------

glm_00 <- lme4::glmer(resp ~ -1 + item + (1|id_i), data = data_stack, family = binomial('logit'), nAGQ = 15)

# ----------------------------------------------- 
# display results
# -----------------------------------------------

broom.mixed::tidy(glm_00) %>%
knitr::kable(., digits = 2)

effect	group	term	estimate	std.error	statistic	p.value
fixed		itemi01	1.69	0.35	4.82	0.00
fixed		itemi02	1.51	0.34	4.41	0.00
fixed		itemi03	0.94	0.32	2.92	0.00
fixed		itemi04	1.42	0.34	4.20	0.00
fixed		itemi05	0.57	0.31	1.82	0.07
fixed		itemi06	0.29	0.31	0.92	0.36
fixed		itemi07	0.22	0.31	0.70	0.48
fixed		itemi08	-0.27	0.31	-0.87	0.38
fixed		itemi09	-0.56	0.31	-1.77	0.08
fixed		itemi10	-0.34	0.31	-1.10	0.27
fixed		itemi11	-0.70	0.32	-2.22	0.03
fixed		itemi12	-2.33	0.40	-5.89	0.00
fixed		itemi13	-1.50	0.34	-4.38	0.00
ran_pars	id_i	sd__(Intercept)	1.43

4 Item bias as mixed model

4.1 Fit Rasch model as a mixed model

\[ln[Pr(y_{ip} = 1)] = \theta_{.p} - \delta_{1-13}item\]

#--------------------------------------------------------------------
# fit rasch model
#--------------------------------------------------------------------

glm_00 <- lme4::glmer(resp ~ -1 + item + (1|id_i), data = data_stack, family = binomial('logit'), nAGQ = 15)

# ----------------------------------------------- 
# display results
# -----------------------------------------------

broom.mixed::tidy(glm_00) %>%
knitr::kable(., digits = 2)

effect	group	term	estimate	std.error	statistic	p.value
fixed		itemi01	1.69	0.35	4.82	0.00
fixed		itemi02	1.51	0.34	4.41	0.00
fixed		itemi03	0.94	0.32	2.92	0.00
fixed		itemi04	1.42	0.34	4.20	0.00
fixed		itemi05	0.57	0.31	1.82	0.07
fixed		itemi06	0.29	0.31	0.92	0.36
fixed		itemi07	0.22	0.31	0.70	0.48
fixed		itemi08	-0.27	0.31	-0.87	0.38
fixed		itemi09	-0.56	0.31	-1.77	0.08
fixed		itemi10	-0.34	0.31	-1.10	0.27
fixed		itemi11	-0.70	0.32	-2.22	0.03
fixed		itemi12	-2.33	0.40	-5.89	0.00
fixed		itemi13	-1.50	0.34	-4.38	0.00
ran_pars	id_i	sd__(Intercept)	1.43

4.2 Fit Rasch model as a latent regression

\[ln[Pr(y_{ip} = 1)] = \theta_{.p} - \delta_{1-13}item + \beta_{.p}u_{.p}\]

#--------------------------------------------------------------------
# latent regression main effect
#--------------------------------------------------------------------

glm_01 <- lme4::glmer(resp ~ -1 + item + u + (1|id_i), data = data_stack, family = binomial('logit'), nAGQ = 15)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.0047381 (tol = 0.002, component 1)

# ----------------------------------------------- 
# display results
# -----------------------------------------------

broom.mixed::tidy(glm_01) %>%
knitr::kable(., digits = 2)

effect	group	term	estimate	std.error	statistic	p.value
fixed		itemi01	1.47	0.36	4.12	0.00
fixed		itemi02	1.29	0.35	3.70	0.00
fixed		itemi03	0.72	0.33	2.19	0.03
fixed		itemi04	1.20	0.35	3.48	0.00
fixed		itemi05	0.35	0.32	1.10	0.27
fixed		itemi06	0.07	0.32	0.22	0.83
fixed		itemi07	0.00	0.32	0.00	1.00
fixed		itemi08	-0.49	0.32	-1.52	0.13
fixed		itemi09	-0.77	0.33	-2.38	0.02
fixed		itemi10	-0.56	0.32	-1.73	0.08
fixed		itemi11	-0.92	0.33	-2.80	0.01
fixed		itemi12	-2.55	0.41	-6.26	0.00
fixed		itemi13	-1.72	0.36	-4.84	0.00
fixed		u	0.97	0.42	2.33	0.02
ran_pars	id_i	sd__(Intercept)	1.37

4.3 Fit Rasch model as a latent regression including the DIF effects

\[ln[Pr(y_{ip} = 1)] = \theta_{.p} - \delta_{1-13}item + \beta_{.p}u_{.p} + \gamma_{ip}(item_{i.}u_{.p})\]

#--------------------------------------------------------------------
# model with bias
#--------------------------------------------------------------------

glm_02 <- lme4::glmer(resp ~ -1 + item + u + item:u + (1|id_i), data = data_stack, family = binomial('logit'), nAGQ = 15)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.0445554 (tol = 0.002, component 1)

# ----------------------------------------------- 
# display results
# -----------------------------------------------

broom.mixed::tidy(glm_02) %>%
knitr::kable(., digits = 2)

effect	group	term	estimate	std.error	statistic	p.value
fixed		itemi01	1.80	0.41	4.44	0.00
fixed		itemi02	1.34	0.38	3.52	0.00
fixed		itemi03	0.83	0.36	2.29	0.02
fixed		itemi04	1.24	0.38	3.29	0.00
fixed		itemi05	0.55	0.36	1.55	0.12
fixed		itemi06	0.37	0.36	1.03	0.30
fixed		itemi07	0.09	0.35	0.26	0.79
fixed		itemi08	-0.54	0.36	-1.51	0.13
fixed		itemi09	-0.64	0.36	-1.78	0.08
fixed		itemi10	-0.84	0.37	-2.29	0.02
fixed		itemi11	-1.14	0.38	-3.04	0.00
fixed		itemi12	-3.24	0.56	-5.78	0.00
fixed		itemi13	-2.43	0.46	-5.28	0.00
fixed		u	-0.48	0.81	-0.60	0.55
fixed		itemi02:u	1.23	1.05	1.18	0.24
fixed		itemi03:u	0.91	0.98	0.94	0.35
fixed		itemi04:u	1.33	1.04	1.28	0.20
fixed		itemi05:u	0.53	0.95	0.55	0.58
fixed		itemi06:u	0.10	0.94	0.11	0.92
fixed		itemi07:u	0.99	0.95	1.04	0.30
fixed		itemi08:u	1.62	0.95	1.70	0.09
fixed		itemi09:u	0.81	0.95	0.85	0.40
fixed		itemi10:u	2.58	0.98	2.63	0.01
fixed		itemi11:u	2.23	0.96	2.31	0.02
fixed		itemi12:u	3.10	1.05	2.94	0.00
fixed		itemi13:u	3.51	1.00	3.50	0.00
ran_pars	id_i	sd__(Intercept)	1.44

4.4 Compare fitted models

#--------------------------------------------------------------------
# model comparison
#--------------------------------------------------------------------

# ----------------------------------------------- 
# variables names
# -----------------------------------------------

texreg::screenreg(list(glm_00, glm_01, glm_02), 
  star.symbol = "\\*",
  single.row = TRUE, 
  center = TRUE, 
  doctype = FALSE,
  dcolumn = TRUE, 
  booktabs = TRUE)

## 
## ==========================================================================================
##                        Model 1                Model 2                Model 3              
## ------------------------------------------------------------------------------------------
## itemi01                   1.69 (0.35) \*\*\*     1.47 (0.36) \*\*\*     1.80 (0.41) \*\*\*
## itemi02                   1.51 (0.34) \*\*\*     1.29 (0.35) \*\*\*     1.34 (0.38) \*\*\*
## itemi03                   0.94 (0.32) \*\*       0.72 (0.33) \*         0.83 (0.36) \*    
## itemi04                   1.42 (0.34) \*\*\*     1.20 (0.35) \*\*\*     1.24 (0.38) \*\*  
## itemi05                   0.57 (0.31)            0.35 (0.32)            0.55 (0.36)       
## itemi06                   0.29 (0.31)            0.07 (0.32)            0.37 (0.36)       
## itemi07                   0.22 (0.31)           -0.00 (0.32)            0.09 (0.35)       
## itemi08                  -0.27 (0.31)           -0.49 (0.32)           -0.54 (0.36)       
## itemi09                  -0.56 (0.31)           -0.77 (0.33) \*        -0.64 (0.36)       
## itemi10                  -0.34 (0.31)           -0.56 (0.32)           -0.84 (0.37) \*    
## itemi11                  -0.70 (0.32) \*        -0.92 (0.33) \*\*      -1.14 (0.38) \*\*  
## itemi12                  -2.33 (0.40) \*\*\*    -2.55 (0.41) \*\*\*    -3.24 (0.56) \*\*\*
## itemi13                  -1.50 (0.34) \*\*\*    -1.72 (0.36) \*\*\*    -2.43 (0.46) \*\*\*
## u                                                0.97 (0.42) \*        -0.48 (0.81)       
## itemi02:u                                                               1.23 (1.05)       
## itemi03:u                                                               0.91 (0.98)       
## itemi04:u                                                               1.33 (1.04)       
## itemi05:u                                                               0.53 (0.95)       
## itemi06:u                                                               0.10 (0.94)       
## itemi07:u                                                               0.99 (0.95)       
## itemi08:u                                                               1.62 (0.95)       
## itemi09:u                                                               0.81 (0.95)       
## itemi10:u                                                               2.58 (0.98) \*\*  
## itemi11:u                                                               2.23 (0.96) \*    
## itemi12:u                                                               3.10 (1.05) \*\*  
## itemi13:u                                                               3.51 (1.00) \*\*\*
## ------------------------------------------------------------------------------------------
## AIC                    1156.68                1153.37                1146.94              
## BIC                    1225.94                1227.57                1280.50              
## Log Likelihood         -564.34                -561.68                -546.47              
## Num. obs.              1040                   1040                   1040                 
## Num. groups: id_i        80                     80                     80                 
## Var: id_i (Intercept)     2.05                   1.89                   2.07              
## ==========================================================================================
## \*\*\* p < 0.001; \*\* p < 0.01; \* p < 0.05

4.5 Commented output


=================================================================================
                       Model 1             Model 2             Model 3           
---------------------------------------------------------------------------------
itemi01                   1.69 (0.35) ***     1.47 (0.36) ***     1.80 (0.41) *** # these are deltas
itemi02                   1.51 (0.34) ***     1.29 (0.35) ***     1.34 (0.38) ***
itemi03                   0.94 (0.32) **      0.72 (0.33) *       0.83 (0.36) *  
itemi04                   1.42 (0.34) ***     1.20 (0.35) ***     1.24 (0.38) ** 
itemi05                   0.57 (0.31)         0.35 (0.32)         0.55 (0.36)    
itemi06                   0.29 (0.31)         0.07 (0.32)         0.37 (0.36)    
itemi07                   0.22 (0.31)        -0.00 (0.32)         0.09 (0.35)    
itemi08                  -0.27 (0.31)        -0.49 (0.32)        -0.54 (0.36)    
itemi09                  -0.56 (0.31)        -0.77 (0.33) *      -0.64 (0.36)    
itemi10                  -0.34 (0.31)        -0.56 (0.32)        -0.84 (0.37) *  
itemi11                  -0.70 (0.32) *      -0.92 (0.33) **     -1.14 (0.38) ** 
itemi12                  -2.33 (0.40) ***    -2.55 (0.41) ***    -3.24 (0.56) ***
itemi13                  -1.50 (0.34) ***    -1.72 (0.36) ***    -2.43 (0.46) ***
u                                             0.97 (0.42) *      -0.48 (0.81)     # <- measure of impact
itemi02:u                                                         1.23 (1.05)     #    this is beta
itemi03:u                                                         0.91 (0.98)    
itemi04:u                                                         1.33 (1.04)    
itemi05:u                                                         0.53 (0.95)    
itemi06:u                                                         0.10 (0.94)    
itemi07:u                                                         0.99 (0.95)    
itemi08:u                                                         1.62 (0.95)    
itemi09:u                                                         0.81 (0.95)    
itemi10:u                                                         2.58 (0.98) **  # items with diff
itemi11:u                                                         2.23 (0.96) *   # these are gamma
itemi12:u                                                         3.10 (1.05) **  #
itemi13:u                                                         3.51 (1.00) *** #
---------------------------------------------------------------------------------
AIC                    1156.68             1153.37             1146.94           
BIC                    1225.94             1227.57             1280.50           
Log Likelihood         -564.34             -561.68             -546.47           
Num. obs.              1040                1040                1040              
Num. groups: id_i        80                  80                  80              
Var: id_i (Intercept)     2.05                1.89                2.07           
=================================================================================
*** p < 0.001, ** p < 0.01, * p < 0.05

Note: when a mixed model is fitted, to retrieve a Rasch Model, the delta parameters are “reverse”. The easier items are “positive”, and the most difficult items are “negative” (i.e. preceded with a “-” sign).

4.6 References

De Boeck, P., Bakker, M., Zwitser, R., Nivard, M., Hofman, A., Tuerlinckx, F., & Partchev, I. (2011). The Estimation of Item Response Models with the lmer Function from the lme4 Package in R. Journal Of Statistical Software, 39(12), 1–28. https://doi.org/10.18637/jss.v039.i12

5 Overall DIF detection

#--------------------------------------------------------------------
# diff detection
#--------------------------------------------------------------------

# ----------------------------------------------- 
# no anchor
# -----------------------------------------------

# Note: DIF is assess without specifying comparable items.

difR::dichoDif(
  Data=data_wide, 
  group="u", 
  focal.name=1, 
  method=c("MH","LRT"),
  model="1PL",
  nrIter=20)



# ----------------------------------------------- 
# with anchor
# -----------------------------------------------

# Note: DIF is assess using items i01-i09 as comparable items

difR::dichoDif(
  Data=data_wide, 
  group="u", 
  focal.name=1, 
  method=c("MH","LRT"),
  model="1PL",
  anchor = 1:9,
  nrIter=20)


# ----------------------------------------------- 
# with purify
# -----------------------------------------------

# Note: DIF is assess letting the library searching for "comparable" items.

difR::dichoDif(
  Data=data_wide, 
  group="u", 
  focal.name=1, 
  method=c("MH","LRT"),
  model="1PL",
  purify=TRUE,
  nrIter=20)

# ----------------------------------------------- 
# shortened version
# -----------------------------------------------

# Note: DIF is assess only among the comparable items.

selected_data <- dplyr::select(raw_data, i01:i09, u)

difR::dichoDif(
  Data=selected_data, 
  group="u", 
  focal.name=1, 
  method=c("MH","LRT"),
  model="1PL",
  purify=TRUE,
  nrIter=20)

5.1 References

Magis, D., Béland, S., Tuerlinckx, F., & de Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42(3), 847–862. https://doi.org/10.3758/BRM.42.3.847

Rasch model with item bias

item bias

dacarras

Mon May 17 16:43:49 2021

0.1 Case Description

1 Cargar datos

1.1 Abrir datos

1.2 Re-structure data

2 Item bias with the logistic method

2.1 Item bias with `difR` with logistic method

2.2 Replicate statistics manually

2.3 Item bias with anchors

3 Item bias as an special case of explanatory response model problem

3.1 Fit Rasch model as a mixed model

4 Item bias as mixed model

4.1 Fit Rasch model as a mixed model

4.2 Fit Rasch model as a latent regression

4.3 Fit Rasch model as a latent regression including the DIF effects

4.4 Compare fitted models

4.5 Commented output

4.6 References

5 Overall DIF detection

5.1 References

Rasch model with item bias

item bias

dacarras

Mon May 17 16:43:49 2021

0.1 Case Description

1 Cargar datos

1.1 Abrir datos

1.2 Re-structure data

2 Item bias with the logistic method

2.1 Item bias with difR with logistic method

2.2 Replicate statistics manually

2.3 Item bias with anchors

3 Item bias as an special case of explanatory response model problem

3.1 Fit Rasch model as a mixed model

4 Item bias as mixed model

4.1 Fit Rasch model as a mixed model

4.2 Fit Rasch model as a latent regression

4.3 Fit Rasch model as a latent regression including the DIF effects

4.4 Compare fitted models

4.5 Commented output

4.6 References

5 Overall DIF detection

5.1 References

2.1 Item bias with `difR` with logistic method